Aragorn lies on a stone bed and is entombed, Arwen lays herself on the grass and dies alone- even though they are together they eventually die among their own lands if not their own people. As to how Arwen was buried it's likely she did inform Eldarion or her daughters she was coming back to Lothlorien- otherwise someone would not know that most of Lorien's people were gone! Yet there is no marker or tomb for her (Gondor style) so Eldarion might have contacted his relatives in Greenwood for help, maybe even Legolas'' group in Ithilien once he had confirmed his mother was dead. I cannot imagine Arwen not leaving behind some instructions for her son and heir.